Find all values of $k$ such that $x^2+x+1$ is a factor of $x^{2k}+x^k+1$.

Question

Find all values of $k$ such that $x^2+x+1$ is a factor of $x^{2k}+x^k+1$. I tried treating the first polynomial as a root of the other but didn’t get anywhere :(. I also tried substitution to get the second polynomial the resemble the first one but also didn’t get anywhere. Just $k=1$?

Answer 1

Let $\omega$ be a root of $x^2+x+1$, then write $$x^{2k}+x^k+1=(x^k)^2+(x^k)+1$$

The task has now been reduced to find out for which $k$ is $\omega^k$ a root of $$x^2+x+1$$

Hint:

What is the value of $\omega^3$?